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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '14400.bm', 'ambient_counter': 39, 'ambient_order': 14400, 'ambient_tex': 'F_5^2:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 900, 'counter': 67, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '14400.bm.8.s1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '8.s1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 8, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '1800.655', 'subgroup_hash': 655, 'subgroup_order': 1800, 'subgroup_tex': 'C_{15}^2:D_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '14400.bm', 'aut_centralizer_order': None, 'aut_label': '8.s1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '14400.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.h1.b1', '4.i1.a1', '4.o1.a1'], 'contains': ['16.c1.a1', '16.o1.b1', '16.p1.a1', '24.bc1.b1', '24.bg1.b1', '200.o1.b1'], 'core': '16.c1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [3001, -1, 1762, -1, 1994, -1, 1971, -1], 'generators': [1, 3456, 6936, 4, 1446, 2880, 1920], 'label': '14400.bm.8.s1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.h1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '8.s1.b1', 'projective_image': '14400.bm', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.s1.b1', 'subgroup_fusion': None, 'weyl_group': '7200.bx'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [10, 10, 10, 4, 12, 4], 'aut_gens': [[1, 2, 24, 120], [1101, 1534, 1512, 864], [1481, 454, 1104, 480], [13, 446, 1512, 864], [733, 742, 816, 1392], [1093, 334, 1104, 528], [17, 154, 48, 240]], 'aut_group': None, 'aut_hash': 8917715223066415996, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14400, 'aut_permdeg': 120, 'aut_perms': [1526123090067057620940022205465378125512354878194936948729444689588650631419754395996998064366615659206655222231351433578284103174514203105512966814265842791547555766830240416181658322980470112261955, 2613057115881457647169160431929779444952306929410126588589273024969635181410530987236230950799939196454575192823884797567648538988270760373338624194504692110836870608943386166697816069402730018772054, 1381837434761462577449992108870525263648504661427436736470243907156297086160564290773506897967553428699462574764344253565122812531965449308641453916239246418586846906383733658687934609609648085962056, 6509881664986768027201856309022214734955396189443231051393208086474496241547187319548597866904746516482639411962187916555814385110826219387641117238334135818018555990004515627470561811206376876094297, 5516198179576673533612713664287008468854103810460743799130103027991022603767871492412935237473859611463049040797544776385197912328205710040588906918356102132133848213972615711900337499434231643514538, 5413942631525570768545420340569569099367676279293582230290510095547640173349051332865715661650173905602543895077413776128105158088136139576014267735998848556808732380212396433334071631984675174689490], 'aut_phi_ratio': 30.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [2, 30, 1, 1], [2, 90, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 150, 1, 1], [5, 4, 2, 2], [5, 8, 1, 1], [6, 30, 2, 1], [6, 50, 1, 2], [6, 100, 1, 1], [10, 60, 2, 1], [10, 180, 2, 1], [12, 150, 2, 1], [15, 4, 4, 1], [15, 8, 2, 5], [15, 8, 4, 3], [30, 60, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^2.C_6^2.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': None, 'autcentquo_hash': 8917715223066415996, 'autcentquo_nilpotent': False, 'autcentquo_order': 14400, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^2.C_6^2.C_2^4', 'cc_stats': [[1, 1, 1], [2, 25, 1], [2, 30, 1], [2, 90, 1], [3, 2, 2], [3, 4, 1], [4, 150, 1], [5, 4, 4], [5, 8, 1], [6, 30, 2], [6, 50, 2], [6, 100, 1], [10, 60, 2], [10, 180, 2], [12, 150, 2], [15, 4, 4], [15, 8, 22], [30, 60, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1800.655', 'commutator_count': 1, 'commutator_label': '450.27', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 655, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 25, 1, 1], [2, 30, 1, 1], [2, 90, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 150, 1, 1], [5, 4, 2, 2], [5, 8, 1, 1], [6, 30, 2, 1], [6, 50, 1, 2], [6, 100, 1, 1], [10, 60, 2, 1], [10, 180, 2, 1], [12, 150, 2, 1], [15, 4, 4, 1], [15, 8, 2, 7], [15, 8, 4, 2], [30, 60, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 60, 'exponents_of_order': [3, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 0, 8], [8, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1800.655', 'hash': 655, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 12, 5, 15], 'inner_gens': [[1, 22, 816, 120], [5, 2, 72, 264], [1129, 74, 24, 120], [1, 1778, 24, 120]], 'inner_hash': 655, 'inner_nilpotent': False, 'inner_order': 1800, 'inner_split': True, 'inner_tex': 'C_{15}^2:D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 9], [4, 18], [8, 23]], 'label': '1800.655', 'linC_count': 32, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 30, 'linQ_dim': 12, 'linQ_dim_count': 30, 'linR_count': 84, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C15^2:D4', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 28, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 54, 'number_divisions': 29, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 133, 'number_subgroup_classes': 140, 'number_subgroups': 2888, 'old_label': None, 'order': 1800, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 145], [3, 8], [4, 150], [5, 24], [6, 260], [10, 480], [12, 300], [15, 192], [30, 240]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 12, 0], 'outer_gens': [[13, 10, 24, 1728], [1, 2, 48, 240], [1, 10, 24, 120]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [4, 4], [8, 5], [16, 9], [32, 2]], 'representations': {'PC': {'code': 7633145960811125796068312976578221737099803680669815, 'gens': [1, 2, 5, 6], 'pres': [7, 2, 2, 2, 3, 5, 3, 5, 309, 36, 422, 58, 451, 28564, 1271, 858, 5556, 5059, 166, 19417, 17660]}, 'Perm': {'d': 16, 'gens': [1327353718456, 1335497143823, 33745299840, 325, 435, 2891867655600, 4311704275200]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^2:D_4', 'transitive_degree': 30, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [8, 12, 8, 20, 8], 'aut_gens': [[1, 2, 12, 48, 2880], [6929, 1942, 13860, 10644, 2304], [5586, 14229, 3468, 14260, 11520], [1769, 8110, 8100, 84, 2304], [6558, 5981, 13260, 11956, 2880], [9042, 10577, 9828, 11416, 2304]], 'aut_group': None, 'aut_hash': 2353156457221532330, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 57600, 'aut_permdeg': 130, 'aut_perms': [2169041547994109583521382196952214171632364798106180382878167494792225253871227941201511405768151510516516257052595999001479473795572437112183950657175080343119133358193141520292090868629554425113197391159297313657636265, 1498133383673661998460371274883244852330742586470764215518840747741790893658161222260352785904970218843590002792599090213334893937911707065893897810016563440851121822042749600946044373467073977190039177691548745714853777, 2180716368273285768046362009340963866106445527919558448076008496985370211723661211661731512595896858326081237261291931525162887068921591588571029094954904337698265396812766113510900802061082559890554363815767844361096366, 1535089113244942985649502841021703120590385442103144430701303943818880801286506248986613693119252075071569662480922296382938394399151650689216294042653617564165498345547113306611300669924745549363011019769817853385026634, 991443897312893931005158269865867896892626037028724811481653558540255422333526782192754699197001823987255133124577719309207145659334839697845820417652168790418462315140751369022440663549769655050884992907466669336240066], 'aut_phi_ratio': 15.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 10, 1, 1], [2, 25, 1, 1], [2, 60, 2, 1], [2, 90, 1, 1], [2, 225, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [4, 10, 1, 2], [4, 25, 1, 2], [4, 50, 1, 3], [4, 90, 1, 2], [4, 225, 1, 2], [4, 300, 2, 1], [4, 450, 1, 3], [5, 8, 1, 1], [5, 16, 1, 1], [6, 20, 2, 2], [6, 50, 2, 1], [6, 100, 1, 1], [6, 120, 2, 1], [8, 300, 2, 2], [10, 40, 1, 1], [10, 72, 1, 1], [10, 144, 1, 1], [10, 240, 2, 1], [10, 360, 1, 1], [12, 20, 2, 4], [12, 50, 2, 2], [12, 100, 1, 2], [12, 100, 2, 6], [12, 600, 2, 1], [15, 16, 2, 2], [15, 32, 2, 1], [15, 64, 1, 1], [20, 40, 1, 2], [20, 360, 1, 2], [24, 600, 2, 2], [30, 80, 2, 2], [30, 480, 2, 1], [60, 80, 2, 4]], 'aut_supersolvable': False, 'aut_tex': 'C_{15}^2.C_2^2.C_2^4.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': None, 'autcentquo_hash': 2353156457221532330, 'autcentquo_nilpotent': False, 'autcentquo_order': 57600, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{15}^2.C_2^2.C_2^4.C_2^2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 10, 1], [2, 25, 1], [2, 60, 2], [2, 90, 1], [2, 225, 1], [3, 2, 2], [3, 4, 1], [4, 10, 2], [4, 25, 2], [4, 50, 3], [4, 90, 2], [4, 225, 2], [4, 300, 2], [4, 450, 3], [5, 8, 1], [5, 16, 1], [6, 20, 4], [6, 50, 2], [6, 100, 1], [6, 120, 2], [8, 300, 4], [10, 40, 1], [10, 72, 1], [10, 144, 1], [10, 240, 2], [10, 360, 1], [12, 20, 8], [12, 50, 4], [12, 100, 14], [12, 600, 2], [15, 16, 4], [15, 32, 2], [15, 64, 1], [20, 40, 2], [20, 360, 2], [24, 600, 4], [30, 80, 4], [30, 480, 2], [60, 80, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '14400.bm', 'commutator_count': 1, 'commutator_label': '900.106', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 39, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 10, 1, 1], [2, 25, 1, 1], [2, 60, 1, 2], [2, 90, 1, 1], [2, 225, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 10, 2, 1], [4, 25, 2, 1], [4, 50, 1, 1], [4, 50, 2, 1], [4, 90, 2, 1], [4, 225, 2, 1], [4, 300, 1, 2], [4, 450, 1, 1], [4, 450, 2, 1], [5, 8, 1, 1], [5, 16, 1, 1], [6, 20, 1, 4], [6, 50, 1, 2], [6, 100, 1, 1], [6, 120, 1, 2], [8, 300, 2, 2], [10, 40, 1, 1], [10, 72, 1, 1], [10, 144, 1, 1], [10, 240, 1, 2], [10, 360, 1, 1], [12, 20, 2, 4], [12, 50, 2, 2], [12, 100, 1, 2], [12, 100, 2, 6], [12, 600, 1, 2], [15, 16, 1, 4], [15, 32, 1, 2], [15, 64, 1, 1], [20, 40, 2, 1], [20, 360, 2, 1], [24, 600, 2, 2], [30, 80, 1, 4], [30, 480, 1, 2], [60, 80, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 13418496, 'exponent': 120, 'exponents_of_order': [6, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[16, 0, 4], [16, 1, 4], [64, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '14400.bm', 'hash': 2366487025886083611, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 120, 'inner_gen_orders': [2, 6, 4, 60, 5], 'inner_gens': [[1, 13834, 4644, 5268, 2304], [3461, 2, 36, 5460, 2304], [4633, 26, 12, 3504, 5760], [6925, 7310, 13836, 48, 5760], [3457, 3458, 11532, 11568, 2880]], 'inner_hash': 2366487025886083611, 'inner_nilpotent': False, 'inner_order': 14400, 'inner_split': True, 'inner_tex': 'F_5^2:S_3^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 28], [4, 28], [8, 8], [16, 20], [32, 4], [64, 1]], 'label': '14400.bm', 'linC_count': 640, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 80, 'linQ_dim': 12, 'linQ_dim_count': 80, 'linR_count': 80, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': None, 'name': 'F5^2:S3^2', 'ngens': 10, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 70, 'number_characteristic_subgroups': 37, 'number_conjugacy_classes': 105, 'number_divisions': 77, 'number_normal_subgroups': 77, 'number_subgroup_autclasses': 817, 'number_subgroup_classes': 1228, 'number_subgroups': 58576, 'old_label': None, 'order': 14400, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 479], [3, 8], [4, 2800], [5, 24], [6, 520], [8, 1200], [10, 1096], [12, 2960], [15, 192], [20, 800], [24, 2400], [30, 1280], [60, 640]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[6918, 11329, 12, 1012, 2880], [1, 10, 12, 48, 2880]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 16], [8, 14], [16, 14], [32, 8], [64, 1]], 'representations': {'PC': {'code': '45982040952812607074531310365322676622144008265903406400102158438876046356921218668396457789178894544308703916183410791971244229228616775981525211562015', 'gens': [1, 2, 4, 6, 10], 'pres': [10, 2, 2, 3, 2, 2, 2, 2, 3, 5, 5, 276681, 51, 242, 185763, 733, 113, 692404, 316085, 163815, 17555, 27525, 175, 210006, 158776, 57716, 38966, 206, 844807, 384017, 16677, 2607, 317, 1036808, 518418, 12998, 8688, 230409, 115219, 48039, 48049, 12059, 12069]}, 'Perm': {'d': 16, 'gens': [2810356900284, 1321576381576, 2083697550]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'F_5^2:S_3^2', 'transitive_degree': 30, 'wreath_data': None, 'wreath_product': False}
